# Kinetic Model of Urea-Related Deposit Reactions

^{*}

## Abstract

**:**

## 1. Introduction

_{x}) are one of the main harmful emissions from diesel engines, which has caused great harm to human health, the ecological environment and the climate. Nowadays, selective catalytic reduction (SCR) systems [1] have been increasingly used in diesel as the mainstream device to deal with NO

_{x}emissions [2,3].

_{3}after evaporation, pyrolysis and hydrolysis, and NH

_{3}converts the harmful NO

_{x}into harmless N

_{2}and H

_{2}O under the action of a catalyst. Numerous studies [4,5,6,7,8] have revealed that the exhaust pipe wall of diesel engines with urea-SCR systems was prone to form deposits consisting of undecomposed urea, biuret, and cyanuric acid (CYA). The deposits easily lead to partial or even total blockage of the exhaust pipe, which increases the exhaust back pressure and seriously affects the performance of diesel engines [9].

_{3}and HNCO, and the residual solid product was CYA. In addition, Stradella [14], Carp [15] and Lundström [16] have conducted studies related to urea pyrolysis as well.

## 2. Results

#### 2.1. Solving the Activation Energy of Urea Pyrolysis Reaction

#### 2.1.1. The Flynn–Wall–Ozawa Method

_{12}is the result from two sets of data with heating rates of 5 and 10 °C/min, E

_{23}, E

_{34}and E

_{45}to follow.

_{34}has a large error in the process of calculating the activation energy of urea decomposition. The reason is that the two thermogravimetric curves of heating rates β

_{3}and β

_{4}almost coincide. The difference of temperature T corresponding to them is very small for the same conversion rate α, which brings a large error. Therefore, the set of data was discarded when calculating the total activation energy E. Finally, the activation energy of urea decomposition is 79 kJ/mol (95% confidence interval, CI: 74–83, as shown in Figure A1) calculated from the Ozawa method.

#### 2.1.2. The Friedman-Reich-Levi Method

_{34}has a large error in the process of calculating the activation energy of urea decomposition. The reason is that the two thermogravimetric curves of heating rates β

_{3}and β

_{4}almost coincide. The difference between temperature T and dα/dT corresponding to them is very small for the same conversion rate α, which brings a large error. Therefore, the set of data was discarded when calculating the total activation energy E. Finally, the activation energy of urea decomposition is 80 kJ/mol (95% confidence interval, CI: 68–93, as shown in Figure A2) calculated from the Friedman two-interval method and 84 kJ/mol (95% confidence interval, CI: 66–103, as shown in Figure A3) calculating from the Friedman one-interval method.

#### 2.1.3. The Kissinger–Akahira–Sunose Method

_{p}is obtained at different heating rates. Table 4 shows the result of the activation energy of urea decomposition by substituting these data into Equation (32).

_{34}has a large error in the process of calculating the activation energy of urea decomposition. The reason is that the difference between the peak temperatures T

_{p}on the two DSC curves of heating rates β

_{3}and β

_{4}is very small, which brings a large error. Therefore, the set of data was discarded when calculating the total activation energy E. Finally, the activation energy of urea decomposition is 82 kJ/mol (95% confidence interval, CI: 55–110, as shown in Figure A4) calculated from the Kissinger method.

#### 2.2. Solving the Activation Energy of Cyanuric Acid (CYA) Pyrolysis Reaction

#### 2.2.1. The Flynn–Wall–Ozawa Method

#### 2.2.2. The Friedman–Reich–Levi Method

#### 2.2.3. The Kissinger–Akahira–Sunose Method

_{p}is obtained at different heating rates. Table 8 shows the result of the activation energy of CYA decomposition by substituting these data into Equation (32). Finally, the activation energy of CYA decomposition is 150 kJ/mol (95% confidence interval, CI: 109–190, as shown in Figure A8) calculated from the Kissinger method.

#### 2.3. Kinetic Modeling of Deposit Reaction

#### 2.3.1. Reaction Path

_{4}

^{+}+ NCO

^{−}

_{4}

^{+}→ NH

_{3}+ H

^{+}

^{−}+ H

^{+}→ HNCO

^{−}+ H

^{+}→ Biuret

^{−}+ H

^{+}

^{−}+ H

^{+}→ CYA + NH

_{3}

^{−}+ 3H

^{+}

^{−}+ H

^{+}→ Ammelide + CO

_{2}

^{−}+ 2H

^{+}+ HCN + NH

_{4}

^{+}+ NCO

^{−}

^{−}+ H

^{+}+ H

_{2}O (aq) → NH

_{3}+ CO

_{2}

^{−}+ H

^{+}→ Biuret

#### 2.3.2. Reaction Rate Equation

_{ki}is the stoichiometric coefficient of the component k in the i-step reaction; ${A}_{i}^{\u2019}$ is the reaction pre-exponential factor; E

_{a,i}is the reaction activation energy; Cs

_{j}is the surface concentration of the component j.

_{k}is the active site occupied by component k; W

_{k}is the molecular mass of the component k.

_{j}can be calculated by the following equation:

^{dying}and A

^{dying}are coefficients; f

_{H2O}is the component concentration of water in the aqueous urea solution; ${f}_{{\mathrm{H}}_{2}\mathrm{O}}^{\mathrm{max}}$ takes the value of 0.876.

^{cry}and A

^{cry}are coefficients; C

^{cry}takes the value of 233.4 K.

#### 2.3.3. The Model Parameters Identification and Validation

## 3. Materials and Methods

#### 3.1. Test Equipment

#### 3.2. Test Sample

#### 3.3. Test Conditions

#### 3.4. Kinetic Analysis Method of Thermal Analysis Curves

#### 3.4.1. The Flynn–Wall–Ozawa Method

_{1}, β

_{1}) and (α, T

_{2}, β

_{2}) with the same conversion rate α on the two TG curves of different heating rates β

_{1}and β

_{2}are substituted into Equation (21) to obtain:

#### 3.4.2. The Friedman–Reich–Levi Method

_{1}, (dα/dT)

_{1}, β

_{1}) and (α, T

_{2}, (dα/dT)

_{2}, β

_{2}) with the same conversion rate α on the two TG curves of different heating rates β

_{1}and β

_{2}are substituted into Equation (25) to obtain:

_{1}, T

_{1}), (α

_{2}, T

_{2}) and (α

_{3}, T

_{3}) when processing the experimental data. For point (α

_{2}, T

_{2}), there are:

- (i)
- Two-interval calculation method: dα/dT ≈ Δα/ΔT = (α
_{3}− α_{1})/(T_{3}− T_{1}); - (ii)
- One-interval calculation method: The points (α
_{1}, T_{1}) and (α_{3}, T_{3}) are averaged to obtain the new point (α′, T′), then dα/dT ≈ Δα/ΔT = (α_{2}− α′)/(T_{2}− T′).

#### 3.4.3. The Kissinger–Akahira–Sunose Method

_{p}

_{1}, β

_{1}) and (T

_{p}

_{2}, β

_{2}) at the peak temperature T

_{p}on the two DSC curves of different heating rates β

_{1}and β

_{2}are substituted into Equation (29) to obtain:

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## Appendix A

## References

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**Figure 1.**Experimental and simulation results comparison of urea pyrolysis (

**a**) Comparison result of Ebrahimian model (

**b**) Comparison result of Brack model.

**Figure 6.**Comparison of the simulation and experiment results of the deposit reaction kinetic model (

**a**) Model in this paper (

**b**) Ebrahimian model. (Solid line-simulation, symbol-test).

1 − α | E_{12} | E_{23} | E_{34} | E_{45} | $\overline{\mathit{E}}$ | E_{O} |
---|---|---|---|---|---|---|

0.95 | 93 | 80 | −597 | 122 | 98 | 79 |

0.90 | 88 | 73 | −632 | 75 | 79 | |

0.85 | 84 | 70 | −625 | 67 | 74 | |

0.80 | 82 | 69 | −764 | 65 | 72 | |

0.75 | 81 | 68 | −1528 | 66 | 72 | |

0.70 | 80 | 68 | −7325 | 70 | 73 | |

0.65 | 81 | 70 | 1484 | 69 | 73 | |

0.60 | 83 | 70 | 657 | 75 | 76 | |

0.55 | 85 | 68 | 511 | 79 | 77 | |

0.50 | 87 | 68 | 5239 | 72 | 76 | |

0.45 | 103 | 77 | 514 | 74 | 85 | |

0.40 | 110 | 80 | 420 | 75 | 88 |

**Table 2.**The activation energy E

_{F,2}of urea decomposition from two-interval method (Unit: kJ/mol).

1 − α | E_{12} | E_{23} | E_{34} | E_{45} | $\overline{\mathit{E}}$ | E_{F,2} |
---|---|---|---|---|---|---|

0.95 | 93 | 50 | −539 | 53 | 65 | 80 |

0.90 | 79 | 64 | −696 | 32 | 58 | |

0.85 | 72 | 54 | −664 | 46 | 57 | |

0.80 | 69 | 60 | −736 | 56 | 62 | |

0.75 | 72 | 58 | −1416 | 77 | 69 | |

0.70 | 77 | 68 | −6316 | 74 | 73 | |

0.65 | 86 | 75 | 912 | 33 | 65 | |

0.60 | 89 | 57 | 482 | 174 | 107 | |

0.55 | 91 | 55 | 388 | 181 | 109 | |

0.50 | 103 | 71 | 7189 | 48 | 74 | |

0.45 | 184 | 122 | 260 | 84 | 130 | |

0.40 | 120 | 77 | 494 | 82 | 93 |

**Table 3.**The activation energy E

_{F,1}of urea decomposition from one-interval method (Unit: kJ/mol).

1 − α | E_{12} | E_{23} | E_{34} | E_{45} | $\overline{\mathit{E}}$ | E_{F,1} |
---|---|---|---|---|---|---|

0.95 | 79 | 70 | −432 | 67 | 72 | 84 |

0.90 | 56 | 61 | −696 | 22 | 46 | |

0.85 | 70 | 52 | −858 | 41 | 54 | |

0.80 | 64 | 63 | −1241 | 7 | 45 | |

0.75 | 74 | 79 | −614 | 81 | 78 | |

0.70 | 100 | 63 | −7976 | 60 | 74 | |

0.65 | 90 | 85 | 559 | 85 | 87 | |

0.60 | 76 | 68 | 450 | 37 | 60 | |

0.55 | 101 | 43 | 401 | 16 | 53 | |

0.50 | 105 | 67 | 6323 | 121 | 98 | |

0.45 | 208 | 271 | −1289 | 136 | 205 | |

0.40 | 104 | 234 | −574 | 80 | 139 |

Parameter | Value |
---|---|

E_{12} | 46 |

E_{13} | 51 |

E_{14} | 64 |

E_{15} | 67 |

E_{23} | 64 |

E_{24} | 102 |

E_{25} | 98 |

E_{34} | 431 |

E_{35} | 163 |

E_{45} | 87 |

E_{K} | 82 |

1 − α | E_{12} | E_{23} | E_{34} | $\overline{\mathit{E}}$ | E_{O} |
---|---|---|---|---|---|

0.95 | 149 | 140 | 332 | 207 | 145 |

0.90 | 145 | 144 | 174 | 154 | |

0.85 | 141 | 143 | 152 | 145 | |

0.80 | 138 | 140 | 145 | 141 | |

0.75 | 134 | 137 | 144 | 138 | |

0.70 | 131 | 150 | 132 | 138 | |

0.65 | 132 | 143 | 140 | 138 | |

0.60 | 132 | 144 | 166 | 147 | |

0.55 | 133 | 145 | 162 | 147 | |

0.50 | 134 | 146 | 157 | 146 | |

0.45 | 134 | 148 | 152 | 145 | |

0.40 | 134 | 147 | 149 | 143 | |

0.35 | 134 | 146 | 147 | 142 | |

0.30 | 134 | 149 | 138 | 140 | |

0.25 | 133 | 147 | 137 | 139 | |

0.20 | 133 | 145 | 134 | 137 | |

0.15 | 132 | 143 | 131 | 135 | |

0.10 | 132 | 135 | 131 | 133 |

**Table 6.**The activation energy E

_{F,2}of CYA decomposition from two-interval method (Unit: kJ/mol).

1 − α | E_{12} | E_{23} | E_{34} | $\overline{\mathit{E}}$ | E_{F,2} |
---|---|---|---|---|---|

0.95 | 132 | 146 | 195 | 158 | 138 |

0.90 | 141 | 145 | 123 | 136 | |

0.85 | 127 | 133 | 117 | 126 | |

0.80 | 118 | 134 | 134 | 129 | |

0.75 | 119 | 127 | 144 | 130 | |

0.70 | 122 | 186 | 263 | 190 | |

0.65 | 136 | 142 | 135 | 138 | |

0.60 | 134 | 149 | 237 | 173 | |

0.55 | 132 | 155 | 122 | 136 | |

0.50 | 134 | 148 | 233 | 172 | |

0.45 | 135 | 104 | 164 | 134 | |

0.40 | 130 | 149 | 217 | 165 | |

0.35 | 127 | 81 | 198 | 135 | |

0.30 | 125 | 142 | 94 | 120 | |

0.25 | 130 | 111 | 123 | 121 | |

0.20 | 126 | 123 | 95 | 115 | |

0.15 | 126 | 102 | 82 | 103 | |

0.10 | 132 | 63 | 103 | 99 |

**Table 7.**The activation energy E

_{F,1}of CYA decomposition from one-interval method (Unit: kJ/mol).

1 − α | E_{12} | E_{23} | E_{34} | $\overline{\mathit{E}}$ | E_{F,1} |
---|---|---|---|---|---|

0.95 | 136 | 112 | 401 | 216 | 153 |

0.90 | 141 | 119 | 146 | 135 | |

0.85 | 117 | 134 | 132 | 128 | |

0.80 | 115 | 133 | 121 | 123 | |

0.75 | 110 | 117 | 153 | 127 | |

0.70 | 102 | 220 | 124 | 149 | |

0.65 | 127 | 144 | 123 | 131 | |

0.60 | 130 | 148 | 763 | 347 | |

0.55 | 103 | 180 | 110 | 131 | |

0.50 | 126 | 139 | 119 | 128 | |

0.45 | 131 | 116 | 195 | 147 | |

0.40 | 140 | 141 | 84 | 122 | |

0.35 | 130 | 44 | 280 | 151 | |

0.30 | 121 | 218 | 86 | 142 | |

0.25 | 154 | 88 | 150 | 131 | |

0.20 | 123 | 101 | 136 | 120 | |

0.15 | 114 | 115 | 76 | 102 | |

0.10 | 116 | 121 | 415 | 217 |

Parameter | Value |
---|---|

E_{12} | 109 |

E_{13} | 135 |

E_{14} | 134 |

E_{23} | 218 |

E_{24} | 171 |

E_{34} | 130 |

E_{K} | 150 |

Reaction | Initial | After Identification | ||
---|---|---|---|---|

E (kJ/mol) | A (s^{−1}) | E (kJ/mol) | A (s^{−1}) | |

R1 | 84 | 8.50 × 10^{6} | 84 | 8.71 × 10^{6} |

R2 | 40 | 1.50 × 10^{2} | 40 | 1.91 × 10^{2} |

R3 | 10 | 6.57 × 10^{2} | 10 | 6.30 × 10^{2} |

R4 | 115 | 7.87 × 10^{14} | 100 | 8.01 × 10^{14} |

R5 | 250 | 1.50 × 10^{24} | 243 | 2.28 × 10^{24} |

R6 | 150 | 2.81 × 10^{18} | 144 | 2.83 × 10^{18} |

R7 | 260 | 1.50 × 10^{19} | 152 | 2.50 × 10^{10} |

R8 | 35 | 3.48 × 10^{5} | 36 | 3.35 × 10^{5} |

R9 | 220 | 6.00 × 10^{14} | 212 | 5.67 × 10^{14} |

R10 | 84 | 1.20 × 10^{8} | 84 | 1.20 × 10^{8} |

R11 | 59 | 5.62 × 10^{9} | 59 | 5.62 × 10^{9} |

R12 | 115 | 3.93 × 10^{14} | 115 | 3.93 × 10^{14} |

R13 | Dying_K | Dying_A | Dying_K | Dying_A |

1 | 100 | 1 | 100 | |

R14 | Cry_K | Cry_A | Cry_K | Cry_A |

1 | −0.000 5 | 1 | −0.000 5 |

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**MDPI and ACS Style**

Zhu, N.; Hong, Y.; Qian, F.; Xu, X.
Kinetic Model of Urea-Related Deposit Reactions. *Molecules* **2023**, *28*, 2340.
https://doi.org/10.3390/molecules28052340

**AMA Style**

Zhu N, Hong Y, Qian F, Xu X.
Kinetic Model of Urea-Related Deposit Reactions. *Molecules*. 2023; 28(5):2340.
https://doi.org/10.3390/molecules28052340

**Chicago/Turabian Style**

Zhu, Neng, Yu Hong, Feng Qian, and Xiaowei Xu.
2023. "Kinetic Model of Urea-Related Deposit Reactions" *Molecules* 28, no. 5: 2340.
https://doi.org/10.3390/molecules28052340